- The first motivating application is time-harmonic wave focusing through a random medium.
Wavefront-shaping-based schemes
[1]}, [2]}, [3]}, [4]}, [5]}
have indeed attracted attention in recent years, particularly because of their potential applications
for focusing and imaging through scattering media.
The primary goal is to focus monochromatic light through a layer of strongly scattering material.
This is a challenging problem as multiple scattering of waves scrambles the transmitted light into random interference intensity patterns called speckle patterns [6]}.
This is shown in Figure REF (a): without control of the source the intensity of the transmitted
field forms a complex speckle pattern.
However, by using a spatial light modulator (SLM) before the scattering medium,
it is possible to focus light as first demonstrated in [4]}.
Indeed, the elements of the SLM can impose phase shifts, and an optimization scheme makes it possible to choose the phase shifts so as to maximize the
intensity transmitted at one target point behind the scattering medium. This is shown in Figure REF (b).
The optimal phase shifts depend on the medium and they are equal to the opposite phases of the field emitted by a point source at the target point
and recorded in the plane of the SLM [8]}.
In other words, the wavefront-shaping optimization procedure is equivalent to
phase conjugation or time reversal. This is illustrated in Figure REF  which describes a time-reversal experiment.
A time-reversal experiment consists of two steps and it is based on the use of a special device, a time-reversal mirror (TRM), that is used as an array of receivers in the first step and as an array of sources in the second step.
The first step is described in picture (a): a point source emits
a wave that propagates through a scattering medium and that is recorded by the TRM.
The second step is described in picture (b): the recorded signals are time-reversed and reemitted into the same medium by the TRM, and the reemitted waves then focus at the original source point. At a single frequency this process corresponds to
phase conjugation or reemission of the complex conjugate of the recorded wave field by the TRM;
with some abuse of notation we refer to this process
as time-harmonic time reversal.
It has been shown that the speckle memory effect [9]}, [10]}
allows to focus on a neighboring point close to the original target point [3]}, [4]}, [5]},
which opens the way for the transmission of spatial patterns [14]}, [15]}, [16]}, [17]}.
Indeed, one the main manifestations of the spatial memory effect is the following one:
By applying an appropriate and deterministic spatial phase modulation to the conjugated source field in the second step of the time-reversal experiment (Figure REF (b)) one can achieve that
the focusing (red spot) in the bottom right plot is shifted.
By properly composing such
modulated source fields one can transmit a pattern, see [14]} for a detailed discussion.
A main question we want to address here is whether such speckle memory effects
can be exploited also in the frequency domain.
In fact, we show that it is possible to focus a time-harmonic signal with a different
frequency than the one
of the field recorded by the TRM in Figure REF (a). One can even focus a broadband pulse
and this opens the way to the transmission of short pulses, see
[8]} for experimental verification of the frequency memory effect.
The process then corresponds to using and processing the reference phase-conjugated field
in Figure REF (b) in order to focus coherently time-harmonic waves with slightly shifted frequencies. The reference field or
a `guide star' field may then be used over a frequency band to obtain focusing for pulses.
The theoretical description of such a frequency memory effect has so far been an open question.
In Section  we give a quantitative description of the effect of a frequency shift on refocusing,
which is directly related to the speckle memory effect in the frequency domain. We show that the speckle pattern
is only slightly changed when shifting the frequency so that we can use the same source phase field over
a range of frequencies and still obtain focusing for all frequencies in the band.
A main result presented in Section  is that the width \(\Omega \)  of the frequency
band for which we can use the same recorded and conjugated field at the TRM and still achieve focusing
is determined by the speckle coherence frequency \(\Omega _{\rm spec}\) :
\(\Omega \lesssim \Omega _{\rm spec} := \frac{\ell _{\rm par}}{L T} ,\) 
